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Spinor Sheaves and Complete Intersections of Quadrics

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Spinor Sheaves and Complete Intersections of Quadrics SPINOR SHEAVES AND COMPLETE INTERSECTIONS OF QUADRICS by Nicolas Addington A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) at the UNIVERSITY OF WISCONSIN–MADISON 2009 SPINOR SHEAVES AND COMPLETE INTERSECTIONS OF QUADRICS Nicolas Addington Under the supervision of Associate Professor Andrei C˘ald˘araru At the University of Wisconsin–Madison We show that the bounded derived category of coherent sheaves on a general complete intersection of four quadrics in P2n−1, n ≥ 4, has a semi-orthogonal decomposition hO(−2n + 9),..., O(−1), O, Di, where D is the derived category of twisted sheaves on a certain non-K¨ahler complex 3-fold. To do this, we develop a theory of “spinor sheaves” on singular quadrics, gener- alizing the spinor bundles on smooth quadrics. i Acknowledgements I am pleased to thank my advisor Andrei C˘ald˘araru, who posed this problem to me and taught me the subject, for his patience and generosity; Joel Robbin, Sean Paul, and Jordan Ellenberg for many helpful discussions, Jean-Pierre Rosay for his perspective on writing, and Jack Lee for his guidance; my long-suffering officemates Matt Davis and Dan Turetsky for their continual willingness to discuss half-baked ideas; and finally my wife Rachel Rodman, my father Michael Addington, and Ann McCreery, who is a very competent mother [39]. ii Contents Acknowledgements i 1 Overview 1 1.1 Intersections of Quadrics ........................... 1 1.2 String Theory ................................. 6 1.3 Matrix Factorizations ............................. 8 2 Techniques 11 2.1 Quadrics .................................... 11 2.2 Spinor Bundles ................................ 17 2.3 Stable Sheaves ................................. 23 2.4 Fourier–Mukai Transforms .......................... 27 2.5 Semi-Orthogonal Decompositions ...................... 30 2.6 Matrix Factorizations ............................. 33 2.7 Twisted Sheaves ................................ 37 2.8 Ordinary Double Points ........................... 40 3 Spinor Sheaves 44 3.1 The Construction ............................... 45 3.2 Dependence on the Subspace ......................... 47 3.3 The Dual ................................... 54 3.4 Linear Sections and Cones .......................... 55 iii 3.5 Stability .................................... 57 4 Intersections of Quadrics 61 4.1 Construction of the Pseudo-Universal Bundle and Small Resolution ... 63 4.2 Embedding of D(Mˆ ,α−1) .......................... 73 4.3 Semi-Orthogonal Decomposition of D(X) .................. 76 Bibliography 80 1 Chapter 1 Overview In §1.1 we review earlier results on intersections of quadrics and state the main theorem. In §1.2 we discuss it in the context of string theory. In §1.3 we discuss it in relation to Orlov’s result on matrix factorizations and derived categories of hypersurfaces and indulge in a little speculation. In Chapter 2 we review the main techniques used in this dissertation: linear spaces on quadrics, spinor sheaves, stable sheaves, Fourier–Mukai transforms, semi-orthogonal decompositions, matrix factorizations, twisted sheaves, and ordinary double points. In Chapter 3 we introduce spinor sheaves on singular quadrics and develop the theory we need to work with them. In Chapter 4 we prove the main theorem. 1.1 Intersections of Quadrics In 1954 Weil was looking for evidence of his famous conjectures, and thus wanted to count points on varieties over Fq. To a general complete intersection X of two quadrics in P2n−1 he associated the following variety [51]: let L be the line those quadrics span 2n+1 in the space P( 2 )−1 of quadrics in P2n−1 and M the double cover of L branched over the 2n points of L corresponding to singular quadrics. Then M is a hyperelliptic curve of genus n − 1, on which he knew how to count points, and he was able to relate the 2 number of points on X to the number on M, and thus compute the zeta function of X. Hirzebruch [22] had just computed the Hodge numbers of complete intersections in projective space; the Hodge diamond of X is 1 . 1 0 ··· 0 n − 1 n − 1 0 ··· 0 1 . 1, which contains that of M 1 n − 1 n − 1 1 in the middle, and Weil was able to verify his conjectures in this example. In his 1972 thesis, Reid [48] showed that this is not merely a coincidence of Hodge numbers, but that the weight 1 Hodge structure of M is isomorphic to the weight 2n−3 Hodge structure of X, or equivalently that the Jacobian of M is isomorphic to the intermediate Jacobian of X.∗ He also showed that the Jacobian of M is isomorphic to the Fano variety of Pn−2s on X. Donagi [17] clarified this, showing that the Abel-Jacobi map from this Fano variety to the intermediate Jacobian is an isomorphism. Donagi also observed that X can be recovered from M, which gives a Torelli theorem for X. Thus the moduli space of degree 0 line bundles on M is isomorphic to the variety of ∗Even better, M is a direct summand of X in the category of motives. 3 Pn−2s on X. In a series of papers starting in 1976 [16], Desale and Ramanan described various moduli spaces of bundles on M in terms of varieties of linear spaces on X. In 1995 Bondal and Orlov [6] gave a categorical explanation of this: viewing M as the fine moduli space of spinor bundles on X, they used the universal bundle as the kernel of a Fourier–Mukai transform to embed D(M) in D(X) and showed that D(X)= hOX (−2n + 5),..., OX (−1), OX ,D(M)i. (1.1.1) Thus any moduli space of objects of D(M) is isomorphic to one of objects of D(X). This can also be seen as a refinement of the Hodge-theoretic results mentioned above. Note that if we set n = 2, so X and M are dual elliptic curves, we recover one of Mukai’s original examples [41] of a derived equivalence. Next we consider a general complete intersection X of three quadrics in P2n−1, the associated plane L in the space of quadrics, and the double cover M of L branched over the locus of singular quadrics, which is a smooth curve of degree 2n. Mukai [42] initiated this study in the case n = 3, so X and M are K3 surfaces, describing M as the moduli space of spinor bundles on X. It need not be a fine moduli space, so we only get 2 ∗ a twisted pseudo-universal bundle, twisted by some Brauer class α ∈ H (M, OM). In his thesis on twisted sheaves, C˘ald˘araru [11] showed that the Fourier–Mukai transform with this twisted bundle as kernel is an equivalence D(X) ∼= D(M,α−1). Following Mukai, many authors studied three quadrics when n > 3; to name a few, O’Grady [43] studied the Hodge structure, Desale [5] studied varieties of linear spaces on X as moduli spaces of bundles on M, and Laszlo [34] proved a Torelli theorem. Again 4 M is a (not necessarily fine) moduli space of spinor bundles on X, and −1 D(X)= hOX (−2n + 7),..., OX (−1), OX ,D(M,α )i. (1.1.2) Before considering more than three quadrics, let us mention the situation with fewer than two. For a single quadric Q ⊂ P2n−1, Kapranov [27] showed that D(Q)= hOQ(−2n + 3),..., OQ(−1), OQ,S+,S−i (1.1.3) where S+ and S− are the two spinor bundles on Q. This is analogous to (1.1.1) and (1.1.2): the subcategory hS+,S−i is the derived category of two points, which we can view as the double cover M of a point L in the space of quadrics. Even Be˘ılinson’s description [3] of the derived category of P2n−1 2n−1 D(P )= hOP2n−1 (−2n + 1),..., OP2n−1 (−1), OP2n−1 i fits into the sequence, viewing P2n−1 as the complete intersection of zero quadrics. 2n+1 For more than three quadrics there is a problem: the hypersurface ∆ ⊂ P( 2 )−1 of singular quadrics is singular in codimension 2, so the linear space L must now meet its singular locus, so L ∩ ∆ and M are singular. In particular the derived category of M is unpleasant to work with, so to describe D(X) one can either ignore M or resolve its singularities. Kapranov [28] described D(X) as a quotient of the derived category of modules over a generalized Clifford algebra, analogous to Ber˘ste˘ın–Gelfan’d–Gelfan’d’s description of D(P2n−1) as a quotient of the derived category of modules over an exterior algebra. 5 Bondal and Orlov [7] equipped M with a related sheaf of algebas B, viewed (M, B) as a non-commutative resolution of singularities of M, and stated that D(X)= hOX (−2n +2m + 1),..., OX (−1), OX ,D(B-mod)i when n ≥ m, where m is the number of quadrics. Kuznetsov [31] proved this and more using his homological projective duality. For three quadrics, B is just an Azumaya algebra, so this is equivalent to (1.1.2), but in general B is less tame. We will take a more geometric approach to the complete intersection of four quadrics: rather than taking a non-commutative resolution of M, we take a non-K¨ahler one M→ˆ M. Our resolution is modular; that is, the points of Mˆ parametrize sheaves on X. The smooth points of M parametrize stable sheaves on X, and its singular points, which are ordinary double points, parametrize S-equivalence classes of properly semi-stable sheaves. The points of Mˆ over smooth points of M will parametrize the same stable sheaves, and an exceptional line over an ODP of M will parametrize semi-stable sheaves in the corresponding S-equivalence class.
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